Field.md

Linear SpacesFieldsDefinition: A field is a set FFF of scalars with two operations, addition and multiplication, such that the following axioms hold:
Addition
(A1) a+b=b+aa+b=b+aa+b=b+a for all a,b∈Fa,b \in Fa,b∈F (commutativity of addition)
(A2) (a+b)+c=a+(b+c)(a+b)+c=a+(b+c)(a+b)+c=a+(b+c) for all a,b,c∈Fa,b,c \in Fa,b,c∈F (associativity of addition)
(A3) There exists an element 0∈F0 \in F0∈F such that a+0=aa+0=aa+0=a for all a∈Fa \in Fa∈F (existence of additive identity)
This element is commonly denoted by 0F0_F0F​.
(A4) For every a∈Fa \in Fa∈F there exists an element −a∈F-a \in F−a∈F such that a+(−a)=0Fa+(-a)=0_Fa+(−a)=0F​ (existence of additive inverse)
This element is commonly denoted by (−a)(-a)(−a).
Multiplication
(M1) a⋅b=b⋅aa \cdot b=b \cdot aa⋅b=b⋅a for all a,b∈Fa,b \in Fa,b∈F (commutativity of multiplication)
(M2) (a⋅b)⋅c=a⋅(b⋅c)(a \cdot b) \cdot c = a \cdot (b \cdot c)(a⋅b)⋅c=a⋅(b⋅c) for all a,b,c∈Fa,b,c \in Fa,b,c∈F (associativity of multiplication)
(M3) There exists an element in FFF such that a⋅1F=aa \cdot 1_F = aa⋅1F​=a for all a∈Fa \in Fa∈F (existence of multiplicative identity)
(M4) For every a∈Fa \in Fa∈F except 0F0_F0F​ there exists an element a−1∈Fa^{-1} \in Fa−1∈F such that a⋅a−1=1Fa \cdot a^{-1} = 1_Fa⋅a−1=1F​ (existence of multiplicative inverse)
Distributivity
(D1) a⋅(b+c)=a⋅b+a⋅ca \cdot (b+c) = a \cdot b + a \cdot ca⋅(b+c)=a⋅b+a⋅c for all a,b,c∈Fa,b,c \in Fa,b,c∈F (distributivity of multiplication over addition)
At least two elements must exist in a field, additive identity 0F0_F0F​ and multiplicative identity 1F1_F1F​.
Field examples include R\mathbb{R}R, C\mathbb{C}C, Q\mathbb{Q}Q, Zp\mathbb{Z}_pZp​ where ppp is a prime number. R\mathbb{R}R are the real numbers, C\mathbb{C}C are the complex numbers, Q\mathbb{Q}Q are the rational numbers, Zp\mathbb{Z}_pZp​ are the integers modulo ppp.
In a sense, what I understand from the fields are, mathematical concepts that are used to define the operations on the elements of the vector spaces.
Linear SpacesFn\mathbb{F}^nFnDefinition: The set of all lists of length nnn with elements from a field F\mathbb{F}F is denoted by Fn\mathbb{F}^nFn.
Fn={(x1,x2,…,xn)∣xi∈F,i=1,2,…,n}\mathbb{F}^n = \{(x_1, x_2, \ldots, x_n) \mid x_i \in \mathbb{F}, i=1,2,\ldots,n\}Fn={(x1​,x2​,…,xn​)∣xi​∈F,i=1,2,…,n}Addition and scalar multiplication are defined as follows:
(x1,x2,…,xn)+(y1,y2,…,yn)=(x1+y1,x2+y2,…,xn+yn)(x_1, x_2, \ldots, x_n) + (y_1, y_2, \ldots, y_n) = (x_1+y_1, x_2+y_2, \ldots, x_n+y_n)(x1​,x2​,…,xn​)+(y1​,y2​,…,yn​)=(x1​+y1​,x2​+y2​,…,xn​+yn​)α(x1,x2,…,xn)=(αx1,αx2,…,αxn)\alpha(x_1, x_2, \ldots, x_n) = (\alpha x_1, \alpha x_2, \ldots, \alpha x_n)α(x1​,x2​,…,xn​)=(αx1​,αx2​,…,αxn​)#EE501 - Linear Systems Theory at METU